Calculating Horizontal Distance of Dart Fired by Child on Incline

[ngoldfarb]

Okay here is the problem:
A dart gun is fired while being held horizontally at a height of 1.55 m above ground level, and at rest relative to the ground. The dart from the gun travels a horizontal distance of 7.77 m. A child holds the same gun in a horizontal position while sliding down a 39.7 degree incline at a constant speed of 0.869 m/s. The acceleration of gravity is 9.8 m/s^2.

What horizontal distance "x" will the dart travel if the child fires the gun forward when it is 1.1m above the ground? Answer in units of m.

I tried finding out how fast the boy would be traveling in the x direction by using: sin 39.7 * 0.869 m/s but once I have that information, I don't know what to do with it. thanks.

 

[Gokul43201]

I tried finding out how fast the boy would be traveling in the x direction by using: sin 39.7 * 0.869 m/s but once I have that information, I don't know what to do with it. thanks.

Should you use the sine or cosine to find the x-component of the child's velocity?

From the information provided in the first part, can you determine the muzzle velocity of the dart gun?

 

[m2003]

I would approach this problem by first identifying the relevant variables and equations that can be used to solve for the horizontal distance traveled by the dart. In this case, the variables are the initial height of the dart (1.55 m), the initial velocity of the dart (0 m/s), the acceleration of gravity (9.8 m/s^2), the angle of the incline (39.7 degrees), and the constant speed of the child (0.869 m/s). The equation that relates these variables is the projectile motion equation, which is: x = v0*cos(theta)*t, where x is the horizontal distance, v0 is the initial velocity, theta is the angle of the incline, and t is the time of flight.

To solve for the horizontal distance, we need to find the time of flight. This can be done by using the equation for the vertical distance traveled by the dart, which is: y = v0*sin(theta)*t - 1/2*g*t^2, where y is the vertical distance and g is the acceleration of gravity. Since we know that the dart will land at a height of 1.1 m, we can set y equal to 1.1 m and solve for t.

1.1 m = 0 + 0.869 m/s * sin(39.7 degrees) * t - 1/2 * 9.8 m/s^2 * t^2

Solving for t, we get t = 0.388 seconds. Now, we can plug this value into the horizontal distance equation to find the distance traveled by the dart:

x = 0 + 0.869 m/s * cos(39.7 degrees) * 0.388 seconds = 0.592 m

Therefore, the dart will travel a horizontal distance of approximately 0.592 m when fired by the child from a height of 1.1 m on the incline. It is important to note that this is an ideal calculation and does not take into account any external factors such as air resistance or friction on the incline.